Finite Group Databasebeta

← #29706869

Subgroups — #29706869

Order 15616.

2206 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519802C211yesmaximal, characteristic, complemented
2C2×#16840326C211yesmaximal, characteristic, complemented
3C2×#16839758C211yesmaximal, characteristic, complemented
4C22×#14519697C211yesmaximal, characteristic, complemented
5C2×#16839729C211yesmaximal, characteristic, complemented
6#21446820C214yesmaximal, complemented
7#21446815C214yesmaximal, complemented
8#21445538C211yesmaximal, characteristic, complemented
9C2×C61×#537C211yesmaximal, characteristic, complemented
10C23×D244C2211yescharacteristic
11C2×#14519802—24no—
12C2×#14519802—24no—
13C2×#14519802—24no—
14C22×#11758201C2212yes—
15C22×#11758199C2212yes—
16C22×#11758194C2211yescharacteristic
17C23×#92418C2211yescharacteristic
18#16840326—24no—
19#16840326—24no—
20C2×#14519773—21no—
21C2×#14519766—21no—
22C2×#14519751C2214yes—
23C2×#14519745—22no—
24C2×#14519736C2211yescharacteristic
25#16839758—24no—
26C2×#14519701C2212yescomplemented
27C2×#14519698C2212yescomplemented
28C2×#14519697—24no—
29C2×#14519697C2214yes—
30#16839729—24no—
31C2×C22⋊#92418C2214yescomplemented
32#16839524C2214yescomplemented
33C22×#41×C61C2211yescharacteristic
34C22×#41×C61C2211yescharacteristic
35C61×#537—24no—
36C61×#537—24no—
37C2×C22⋊Q8×C61—21no—
38C2×C22⋊Q8×C61C2212yescomplemented
39C2×#121×C61—21no—
40C2×#121×C61—22no—
41C2×#121×C61C2212yescomplemented
42C22×C4⋊C4×C61C2211yescharacteristic
43C24×D61—21no—
44C22×D244—22no—
45C22×D244—22no—
46C22×D244—22no—
47C22×D244—22no—
48C22×D244—21no—
49C22×D244—22no—
50C22×D244D414yescomplemented
51C22×D244C2312yes—
52C22×C22⋊D61—24no—
53#14519802—232no—
54C2×#11758201—24no—
55C2×#11758201—24no—
56C2×#11758201—24no—
57C2×#11758199—24no—
58C2×#11758199—24no—
59C2×#11758199—24no—
60C2×#11758194—22no—
61C2×#11758194—22no—
62C2×#11758194D414yescomplemented
63C22×#92418—22no—
64C22×#92418—22no—
65C22×#92418—22no—
66C22×#92418C2312yes—
67C22×#92418C2312yes—
68#14519773—28no—
69#14519766—28no—
70#14519751—28no—
71#14519751—28no—
72#14519745—216no—
73#14519736—22no—
74#14519736D414yescomplemented
75C2×C4⋊Dic61C2312yes—
76C2×C22⋊Dic61—21no—
77C2×#11758198—24no—
78C2×C4×Dic61—21no—
79#14519701—24no—
80#14519698—24no—
81#14519697—28no—
82C22⋊#92418—28no—
83C2×#41×C61—22no—
84C2×#41×C61—22no—
85C2×#41×C61—22no—
86C2×#41×C61—22no—
87C2×#41×C61—22no—
88C2×#41×C61—22no—
89C2×#41×C61—22no—
90C2×#41×C61—22no—
91C2×#41×C61—22no—
92C2×#41×C61—22no—
93C2×#41×C61—22no—
94C2×#41×C61D414yescomplemented
95C22×Q8×C61D411yescharacteristic
96C22×Q8×C61C2311yescharacteristic
97C22×D4×C61—21no—
98C22×D4×C61—22no—
99C22×D4×C61D412yes—
100C22×D4×C61D411yescharacteristic
101C22×D4×C61C2311yescharacteristic
102C23×C4×C61—21no—
103C23×C4×C61—21no—
104C23×C4×C61C2312yes—
105C23×C4×C61C2311yescharacteristic
106C22⋊Q8×C61—28no—
107C22⋊Q8×C61—24no—
108C22⋊Q8×C61—24no—
109#121×C61—216no—
110#121×C61—28no—
111#121×C61—24no—
112#121×C61—24no—
113C2×C4⋊C4×C61—22no—
114C2×C4⋊C4×C61—21no—
115C2×C4⋊C4×C61—22no—
116C2×C4⋊C4×C61D414yescomplemented
117C2×C4⋊C4×C61C2312yes—
118C2×C22⋊C4×C61—22no—
119C2×C22⋊C4×C61—22no—
120C24×C61—21no—
121C24×C61C2×D412yes—
122C24×C61C2×D411yescharacteristic
123C23×D61—22no—
124C23×D61—24no—
125C23×D61—42no—
126C23×D61—42no—
127C23×D61—82no—
128C23×D61—82no—
129#41×C61—28no—
130#41×C61—28no—
131#41×C61—44no—
132#41×C61—44no—
133#41×C61—44no—
134#41×C61—44no—
135#41×C61—44no—
136#41×C61—44no—
137C2×Q8×C61—22no—
138C2×Q8×C61—22no—
139C2×Q8×C61—22no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61C2×D412yes—
146C2×D4×C61—24no—
147C2×D4×C61—24no—
148C2×D4×C61—24no—
149C2×D4×C61—24no—
150C2×D4×C61—28no—
151C2×D4×C61—22no—
152C2×D4×C61—22no—
153C2×D4×C61—22no—
154C2×D4×C61—21no—
155C2×D4×C61—21no—
156C2×D4×C61—21no—
157C2×D4×C61—21no—
158C2×D4×C61—21no—
159C2×D4×C61—42no—
160C2×D4×C61—42no—
161C2×D4×C61—84no—
162C2×D4×C61C2×D412yes—
163C22×C4×C61—22no—
164C22×C4×C61—21no—
165C22×C4×C61—24no—
166C22×C4×C61—24no—
167C22×C4×C61—22no—
168C22×C4×C61—22no—
169C22×C4×C61—22no—
170C22×C4×C61—22no—
171C22×C4×C61—22no—
172C22×C4×C61—22no—
173C22×C4×C61—42no—
174C22×C4×C61—42no—
175C22×C4×C61—82no—
176C22×C4×C61C2×D412yes—
177C22×C4×C61C2×D411yescharacteristic
178C22×C4×C61C2×D412yes—
179C22×C4×C61C2×D412yes—
180C22×C4×C61C2×D412yes—
181C22×C4×C61C2×D412yes—
182C22×C4×C61C2×D412yes—
183C22×C4×C61#4112yes—
184C22×C4×C61C2411yescharacteristic
185C2×C22⋊D61—216no—
186C2×C22⋊D61—84no—
187C2×C22⋊D61—84no—
188C22×Dic61—22no—
189C2×D244—28no—
190C2×D244—28no—
191C2×D244—22no—
192C2×D244—22no—
193C2×D244—22no—
194C2×D244—22no—
195C2×D244—24no—
196C2×D244—22no—
197C2×D244—42no—
198C2×D244—44no—
199C2×D244—42no—
200C2×D244—44no—
201C2×D244C2×D414yes—
202C4⋊C4×C61—28no—
203C4⋊C4×C61—24no—
204C4⋊C4×C61—22no—
205C4⋊C4×C61C2×D414yes—
206C22⋊C4×C61—28no—
207C22⋊C4×C61—28no—
208C22⋊Dic61—24no—
209#11758201—216no—
210#11758199—216no—
211#11758198—216no—
212C4⋊Dic61—22no—
213C4⋊Dic61C2×D414yes—
214C4×Dic61—24no—
215#11758194—28no—
216C2×#92418—22no—
217C2×#92418—22no—
218C2×#92418—22no—
219C2×#92418—22no—
220C2×#92418—22no—
221C2×#92418—44no—
222C2×#92418C2×D414yes—
223C23×C61—22no—
224C23×C61—22no—
225C23×C61—22no—
226C23×C61—22no—
227C23×C61—22no—
228C23×C61—22no—
229C23×C61—24no—
230C23×C61—24no—
231C23×C61—41no—
232C23×C61—41no—
233C23×C61—42no—
234C23×C61—42no—
235C23×C61#12012yes—
236C23×C61#12012yes—
237C23×C61#12112yes—
238C23×C61C22×D411yescharacteristic
239C22×D61—24no—
240C22×D61—42no—
241C22×D61—42no—
242C22×D61—44no—
243C22×D61—44no—
244C22×D61—48no—
245C22×D61—88no—
246C22×D61—84no—
247C22×D61—84no—
248C22×D61—88no—
249Q8×C61—22no—
250Q8×C61—22no—
251Q8×C61—41no—
252Q8×C61—41no—
253Q8×C61—44no—
254D4×C61—22no—
255D4×C61—22no—
256D4×C61—48no—
257D4×C61—48no—
258D4×C61—41no—
259D4×C61—41no—
260D4×C61—44no—
261D4×C61—88no—
262C2×C4×C61—22no—
263C2×C4×C61—24no—
264C2×C4×C61—22no—
265C2×C4×C61—22no—
266C2×C4×C61—22no—
267C2×C4×C61—41no—
268C2×C4×C61—41no—
269C2×C4×C61—44no—
270C2×C4×C61—42no—
271C2×C4×C61—42no—
272C2×C4×C61—42no—
273C2×C4×C61—42no—
274C2×C4×C61—48no—
275C2×C4×C61—84no—
276C2×C4×C61#12014yes—
277C2×C4×C61#12012yes—
278C2×C4×C61#12012yes—
279C2×C4×C61#12114yes—
280C2×C4×C61#12114yes—
281C2×C4×C61#12112yes—
282C2×C4×C61#12714yes—
283C2×C4×C61C22×D411yescharacteristic
284C2×C4×C61C22×D411yescharacteristic
285C2×C4×C61C22×D411yescharacteristic
286C2×C4×C61C22×D411yescharacteristic
287C2×C4×C61C22×D411yescharacteristic
288C2×C4×C61C2×#4111yescharacteristic
289C22⋊D61—816no—
290C2×Dic61—24no—
291C2×Dic61—42no—
292C2×Dic61—42no—
293D244—24no—
294D244—24no—
295D244—42no—
296D244—42no—
297D244—48no—
298#92418—24no—
299#92418—42no—
300#23393—611nomaximal, Sylow
301C22×C61—22no—
302C22×C61—22no—
303C22×C61—24no—
304C22×C61—24no—
305C22×C61—42no—
306C22×C61—42no—
307C22×C61—42no—
308C22×C61—42no—
309C22×C61—42no—
310C22×C61—42no—
311C22×C61—44no—
312C22×C61—44no—
313C22×C61—82no—
314C22×C61—82no—
315C22×C61#39214yes—
316C22×C61C2×#12011yescharacteristic
317C22×C61C2×#12011yescharacteristic
318C22×C61C2×#12111yescharacteristic
319C22×C61#57612yes—
320C22×C61#57712yes—
321C2×D61—44no—
322C2×D61—48no—
323C2×D61—816no—
324C2×D61—82no—
325C2×D61—82no—
326C2×D61—84no—
327C4×C61—42no—
328C4×C61—42no—
329C4×C61—44no—
330C4×C61#39214yes—
331C4×C61C2×#12011yescharacteristic
332C4×C61C2×#12111yescharacteristic
333C4×C61C2×#12111yescharacteristic
334C4×C61C2×#12711yescharacteristic
335Dic61—44no—
336C22×AGL(1,ℤ/8)—611no—
337C2×#537D6111yescharacteristic, complemented
338C2×#535—611no—
339C2×#416—611no—
340C22×#102—611no—
341C2×#408—611no—
342#2000—614no—
343#1999—614no—
344#1736—611no—
345C2×C61—24no—
346C2×C61—82no—
347C2×C61—82no—
348C2×C61C2×#39211yescharacteristic
349C2×C61#300811yescharacteristic
350C2×C61#301011yescharacteristic
351C2×C61#301212yes—
352C2×C61#306212yes—
353D61—84no—
354C22×#41C2×D6111yescharacteristic
355C22×#41C2×D6111yescharacteristic
356C23×D4—611no—
357C2×AGL(1,ℤ/8)—1224no—
358C2×AGL(1,ℤ/8)—1224no—
359C2×AGL(1,ℤ/8)—1224no—
360C22×SD16—612no—
361C22×D8—612no—
362C22×M16—611no—
363C23×C8—611no—
364#537—24no—
365#537—24no—
366#535—1224no—
367#535—1224no—
368C2×#127—1221no—
369C2×#124—1221no—
370C2×C22⋊Q8—21no—
371C2×C22⋊Q8C2×D6112yescomplemented
372C2×#121—21no—
373C2×#121—22no—
374C2×#121—614no—
375C2×#121C2×D6112yescomplemented
376C2×#120—1222no—
377C2×#117—611no—
378C22×C4⋊C4C2×D6111yescharacteristic
379#416—1224no—
380C2×#103—612no—
381C2×#102—612no—
382C2×#102—614no—
383C2×#102—1224no—
384#408—1224no—
385C2×C22⋊C8—614no—
386#340—614no—
387C61#2339311yescharacteristic, complemented, Sylow
388C25—1221no—
389C2×#41—22no—
390C2×#41—22no—
391C2×#41—22no—
392C2×#41—22no—
393C2×#41—22no—
394C2×#41—22no—
395C2×#41—22no—
396C2×#41—22no—
397C2×#41—22no—
398C2×#41—22no—
399C2×#41—22no—
400C2×#41C22⋊D6114yescomplemented
401C22×Q8C22⋊D6111yescharacteristic
402C22×Q8C22×D6111yescharacteristic
403C22×D4—21no—
404C22×D4—22no—
405C22×D4—614no—
406C22×D4—612no—
407C22×D4—1224no—
408C22×D4—1222no—
409C22×D4—1222no—
410C22×D4—1222no—
411C22×D4—1222no—
412C22×D4—1222no—
413C22×D4—1221no—
414C22×D4C22⋊D6112yes—
415C22×D4C22⋊D6111yescharacteristic
416C22×D4C22×D6111yescharacteristic
417C23×C4—21no—
418C23×C4—21no—
419C23×C4C22×D6112yes—
420C23×C4C22×D6111yescharacteristic
421AGL(1,ℤ/8)—12232no—
422C2×SD16—1224no—
423C2×SD16—1224no—
424C2×SD16—1224no—
425C2×D8—1224no—
426C2×D8—1224no—
427C2×D8—1224no—
428C2×M16—614no—
429C2×M16—1222no—
430C2×M16—1222no—
431C22×C8—612no—
432C22×C8—612no—
433C22×C8—1222no—
434C22×C8—1222no—
435C22×C8—1222no—
436#127—1228no—
437#124—1228no—
438C22⋊Q8—28no—
439C22⋊Q8—24no—
440C22⋊Q8—24no—
441#121—216no—
442#121—28no—
443#121—24no—
444#121—24no—
445#121—1228no—
446#121—1228no—
447#120—12216no—
448#117—614no—
449#117—1222no—
450C2×C4⋊C4—21no—
451C2×C4⋊C4—22no—
452C2×C4⋊C4—22no—
453C2×C4⋊C4—612no—
454C2×C4⋊C4C22⋊D6114yescomplemented
455C2×C4⋊C4C22×D6112yes—
456C2×C22⋊C4—22no—
457C2×C22⋊C4—22no—
458C2×C22⋊C4—1221no—
459C2×C22⋊C4—1224no—
460C2×C42—1221no—
461#103—1224no—
462#102—1224no—
463#102—1228no—
464C22⋊C8—1228no—
465C24—21no—
466C24—1224no—
467C24—1222no—
468C24—4882no—
469C24—4882no—
470C24—2442no—
471C24—2442no—
472C24C2×C22⋊D6112yes—
473C24C2×C22⋊D6111yescharacteristic
474#41—28no—
475#41—28no—
476#41—44no—
477#41—44no—
478#41—44no—
479#41—44no—
480#41—44no—
481#41—44no—
482C2×Q8—21no—
483C2×Q8—22no—
484C2×Q8—22no—
485C2×Q8—22no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8C2×C22⋊D6112yes—
491C2×D4—24no—
492C2×D4—24no—
493C2×D4—24no—
494C2×D4—24no—
495C2×D4—28no—
496C2×D4—21no—
497C2×D4—21no—
498C2×D4—21no—
499C2×D4—21no—
500C2×D4—21no—
501C2×D4—22no—
502C2×D4—22no—
503C2×D4—22no—
504C2×D4—42no—
505C2×D4—42no—
506C2×D4—84no—
507C2×D4—614no—
508C2×D4—12216no—
509C2×D4—1222no—
510C2×D4—1222no—
511C2×D4—1222no—
512C2×D4—1222no—
513C2×D4—1222no—
514C2×D4—1228no—
515C2×D4—1224no—
516C2×D4—1228no—
517C2×D4—4884no—
518C2×D4—4884no—
519C2×D4—2442no—
520C2×D4—2444no—
521C2×D4—2442no—
522C2×D4—2444no—
523C2×D4C2×C22⋊D6112yes—
524C22×C4—22no—
525C22×C4—21no—
526C22×C4—24no—
527C22×C4—22no—
528C22×C4—22no—
529C22×C4—22no—
530C22×C4—22no—
531C22×C4—22no—
532C22×C4—22no—
533C22×C4—24no—
534C22×C4—42no—
535C22×C4—42no—
536C22×C4—82no—
537C22×C4—1222no—
538C22×C4D4×D6112yes—
539C22×C4D4×D6112yes—
540C22×C4D4×D6112yes—
541C22×C4#1175822412yes—
542C22×C4C2×C22⋊D6112yes—
543C22×C4C2×C22⋊D6112yes—
544C22×C4C2×C22⋊D6112yes—
545C22×C4C2×C22⋊D6111yescharacteristic
546C22×C4C23×D6111yescharacteristic
547SD16—12216no—
548D8—12216no—
549M16—1228no—
550C2×C8—614no—
551C2×C8—1222no—
552C2×C8—1222no—
553C2×C8—1222no—
554C2×C8—1222no—
555C2×C8—1222no—
556C2×C8—2444no—
557C4⋊C4—28no—
558C4⋊C4—22no—
559C4⋊C4—24no—
560C4⋊C4—614no—
561C4⋊C4—1222no—
562C4⋊C4C2×C22⋊D6114yes—
563C22⋊C4—28no—
564C22⋊C4—28no—
565C22⋊C4—12216no—
566C22⋊C4—1224no—
567C42—1224no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—22no—
573C23—22no—
574C23—24no—
575C23—24no—
576C23—41no—
577C23—41no—
578C23—42no—
579C23—42no—
580C23—1224no—
581C23—4884no—
582C23—4888no—
583C23—4884no—
584C23—4888no—
585C23—2442no—
586C23—2442no—
587C23—2444no—
588C23—2444no—
589C23—2448no—
590C23#1451974512yes—
591C23#1451974612yes—
592C23#1451974912yes—
593C23C22×C22⋊D6111yescharacteristic
594Q8—22no—
595Q8—22no—
596Q8—41no—
597Q8—41no—
598Q8—44no—
599D4—22no—
600D4—22no—
601D4—48no—
602D4—48no—
603D4—41no—
604D4—41no—
605D4—44no—
606D4—88no—
607D4—1224no—
608D4—1224no—
609D4—48816no—
610D4—2442no—
611D4—2442no—
612D4—2448no—
613C2×C4—22no—
614C2×C4—24no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—48no—
626C2×C4—84no—
627C2×C4—1224no—
628C2×C4—2442no—
629C2×C4—2442no—
630C2×C4#1451974514yes—
631C2×C4#1451974512yes—
632C2×C4#1451974612yes—
633C2×C4#1451974814yes—
634C2×C4#1451974914yes—
635C2×C4#1451974912yes—
636C2×C4#1451977314yes—
637C2×C4C2×D4×D6111yescharacteristic
638C2×C4C2×D4×D6111yescharacteristic
639C2×C4C2×D4×D6111yescharacteristic
640C2×C4C22×C22⋊D6111yescharacteristic
641C2×C4C22×C22⋊D6111yescharacteristic
642C2×C4C2×#1175822411yescharacteristic
643C8—1224no—
644C8—2442no—
645C22—22no—
646C22—24no—
647C22—24no—
648C22—22no—
649C22—42no—
650C22—42no—
651C22—44no—
652C22—42no—
653C22—42no—
654C22—44no—
655C22—42no—
656C22—42no—
657C22—82no—
658C22—82no—
659C22—48816no—
660C22—4882no—
661C22—4884no—
662C22—4882no—
663C22—2448no—
664C22—2444no—
665C22#1683967014yes—
666C22C2×#1451974511yescharacteristic
667C22C2×#1451974611yescharacteristic
668C22C2×#1451974911yescharacteristic
669C22#1684056512yes—
670C22#1684057012yes—
671C4—42no—
672C4—42no—
673C4—44no—
674C4—2444no—
675C4#1683967014yes—
676C4C2×#1451974511yescharacteristic
677C4C2×#1451974811yescharacteristic
678C4C2×#1451974911yescharacteristic
679C4C2×#1451977311yescharacteristic
680C2—24no—
681C2—82no—
682C2—82no—
683C2—4884no—
684C2C2×#1683967011yescharacteristic
685C2#2145508711yescharacteristic
686C2#2145510511yescharacteristic
687C2#2145512012yes—
688C2#2145550112yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#537C2
socleC23×C61C22×D4
hypercenterC2×#537D61
hypocenterC61#23393
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#537

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C22×#14519802C2semidirect
2. C2×#16840326C2semidirect
3. C2×#16839758C2semidirect
4. C22×#14519697C2semidirect
5. C2×#16839729C2semidirect
6. #21446820C2semidirect
7. #21446815C2semidirect
8. #21445538C2semidirect
9. C2×C61×#537C2semidirect
26. C2×#14519701C22semidirect
27. C2×#14519698C22semidirect
31. C2×C22⋊#92418C22semidirect
32. #16839524C22semidirect
38. C2×C22⋊Q8×C61C22semidirect
41. C2×#121×C61C22semidirect
50. C22×D244D4semidirect
62. C2×#11758194D4semidirect
74. #14519736D4semidirect
94. C2×#41×C61D4semidirect
116. C2×C4⋊C4×C61D4semidirect
337. C2×#537D61semidirect
371. C2×C22⋊Q8C2×D61semidirect
375. C2×#121C2×D61semidirect
387. C61#23393semidirect
400. C2×#41C22⋊D61semidirect
454. C2×C4⋊C4C22⋊D61semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×#145198022
C2×#168403262
C2×#168397582
C22×#145196972
C2×#168397292
#214468202
#214468152
#214455382
C2×C61×#5372
#2339361